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The triangle numbers are 1,3,6,10,etc... The nth triangle number is formed by the following sum: $T_n=\sum_{i=1}^n i=1+2+3+...+n$. I want to try and derive the general formula for the nth triangle number in the following manner.

Form a rectangle from squares with the shorter side being n units long and the longer side being n+1 units long. The area of the rectangle will be $n(n+1)$. Now cut the rectangle along one of its diagonals while following the path of the boundaries of the squares, which should give you a staircase shape. Since the rectangle has been effectively cut in half, you have essentially made a "triangle". Sure the shape isn't truly triangular, but the amount of squares that make it up is the same as the nth triangular number, since you take a side away from the longer side of the rectangle. The amount of squares that make it up can then be formulated as $n(n+1)/2$, which is the general formula if you look it up.

Is this sound logic?

X Stanton
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1 Answers1

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It looks like good logic and the proof does work. HOWEVER, you miss the part where you explain why this half of the rectangle means you calculated $1 + 2 + \cdots + n$.

For example, you could just say: "The first row in this "triangle" has $n$ squares, the second has $n-1$ squares... the $n$th row has $1$ square. Thus, the area of this "triangle" is $1 + 2 + \cdots + n$ which we just calculated to be $\frac{n(n+1)}{2}.$

mathz2003
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